130 Mr. Meikle on the 



the quotient is the acceleration of each particle, otherwise ex- 

 pressed by — -— ^ * ; wherefore 



ddz -_ j£ F ddz_ „ 

 dT2 3 * y ' dx"- 



Were every thing correct about this equation and the mode 

 by which Mr. Ivory has obtained it, the velocity would obvi- 

 ously, as he in effect states it, be 



dx __ /Ip'" 



and since both dx and dr are constant, the velocity would be 

 uniform, and always the same in air of the same density and 

 pressure. But another notable error and inconsistency have 

 here evaded notice, by the manoeuvre of twice rejecting the 

 higher powers of dz^ seemingly for the purpose of rendering the 

 calculus manageable, though, as we shall presently see, there 

 was no call or necessity for it on that account. Whether M. 

 Laplace or Mr. Ivory were aware of this circumstance, I could 

 not pretend to say ; but one thing is certain, that further de- 

 fects of the investigation become sufficiently apparent, when 

 none of these powers have been discarded. For in this way 

 we have 



Z. = ( dx \ 



F \ dx+dz ) 



dx+dz 



* Viz. one of the usual differential expressions for an accelerating force. 

 The second fluxion of the space being ddz, and the undefined symbol dr 

 denoting the constant fluxion of the time. It is from this step that it 

 becomes more particularly obvious that the length of the cylinder is a 

 measure of its velocity, being always equal to the minute space described 

 during the constant moment of time dr. Not the shadow of a reason is 

 either given or supposed necessary to assign why the length of the cylin- 

 der should not rather have had some other relation to its velocity than 

 that just mentioned, which we have already seen to be impossible. But 

 the gratuitous assumptions in this investigation are so numerous and im- 

 portant that they would have rendered it null and void as a mathematical 

 production, although no inconsistency had presented itself. For were 

 such assumptions to be tolerated in mathematics, there is no problem, 

 however difficult, but they could solve with the utmost facility. A curious 

 instance of their irresistible powers is noticed in the Phil. Mag. for Dec. 

 1822, where I have shown that the demonstration which Mr. Ivory sup- 

 posed he had given of Euclid's 12th Axiom, in the number for March 

 preceding, owes all its virtue to an assumption fully equivalent to tliQ 

 axiom itself, which was the very point to be proved !— H, M. 



